Jordán, F. 2003. Quantifying landscape connectivity: key patches and key corridors. In: Tiezzi, E., Brebbia, C. A. and Usó, J.-L. (eds.), Ecosystems and Sustainable Development IV, WIT Press, Southampton, pp. 883-892.
Quantifying landscape connectivity: key patches and key corridors
Ferenc Jordán
Department of Plant Taxonomy and Ecology, Eötvös University, Budapest, Hungary
Abstract
The problems of habitat loss and fragmentation call for understanding how do ecological landscapes serve the needs of inhabiting metapopulations. Questions of extreme importance are (1) how connected are landscape graphs in nature, (2) how to measure this connectivity mathematically, and (3) how do the spatial elements of the landscape contribute to the maintenance of connectivity. I propose a landscape graph analysis where the positional importance of individual habitat patches (nodes) and ecological corridors (links) is quantified. This refers to the role of these spatial elements in maintaining landscape connectivity, i.e. how will the landscape graph be damaged after losing its patch or corridor in question. Our approach takes into account both structure and function; functional aspects are expressed by the assessed quality of patches and corridors. I illustrate this approach on a case study and, within the limits of our theory, I set conservation priorities. In other words, I determine which patch’s lost is the worst and which patch’s lost is the least wrong. However this analysis has a single species in focus, I believe that, if adequate species are targeted (keystones, flagships, umbrellas, etc…), the value of single-species approaches may be high. I emphasize that (1) the use of spatial elements can be assessed only in a landscape context, (2) functional aspects must complement pure structural descriptions, and (3) quantitativity is a primary property of effective conservation methods.
1 Introduction
The loss and fragmentation of natural habitats raises the problem whether local populations in small habitat fragments become perfectly isolated or remain connected to others. Small, isolated, local populations face a high risk of extinction, because of genetic, demographic and stochastic reasons. Migration and the resulting gene flow between habitat fragments may well be the only key for survival in the case of a number of species [1]. If corridors connect habitat patches and individuals can migrate through the network of these landscape elements, metapopulation dynamics emerges [2]. The efficiency of migration (and gene flow) depends on the topology of landscape elements (i.e. corridors and patches, cf. Pickett and Cadenasso [3]) and their topographical properties (e.g. corridor length). One of the major attributes of landscape graphs is their connectivity but presently there is no consensus on how to measure it (moreover, certain measures produce the contraintuitive artefact that fragmentation increases connectivity). Further, there is no good approach to how to evaluate the importance of individual landscape elements in maintaining landscape connectivity (against fragmentation). I support the view that the value of either habitat patches or ecological corridors can be assessed only within a network context (cf. van der Sluis and Chardon [4]): a patch described by not very good parameters (e.g. fertility, humidity, local population size) still can be a key habitat fragment if its positionality within the whole landscape graph is extremely important for maintaining migration patterns. However I only target to analyse the landscape structure of a single species, and I acknowledge the limitations of such results, I also hope that using these methods for analysing focal species (e.g. umbrella, flagship, or keystone species) might be of outstanding importance [5].
Here, my aim is to present a considerably small-scale case study for illustrating the use of some graph properties referring to landscape connectivity, to construct a combined importance index for individual spatial elements and to set conservation priorities, within the limits of this approach.
2 Area and the focal species
The seriously endangered, brachypterous bush-cricket Pholidoptera transsylvanica inhabits only tall-grass semi-dry swards. Its predatory behaviour and high mobility ensures that individuals migrate easily between certain patches of this habitat type, if permeable corridors exist between them. The network-like habitat structure of P. transsylvanica is characteristic for the species. The metapopulation inhabiting the Aggtelek Karst region of NE-Hungary is perfectly isolated from the two other Hungarian metapopulations [6, 7], so, individuals live in a well-defined network of suitable habitats.
3 Methods of analysis
3.1 Landscape graph construction
First, the habitat structure of P. transsylvanica was determined, based on field observations (cca in a 40 years interval) and recently detected acoustic signals of stridulating males. Three independent observers proposed nearly exactly the same landscape graph, and, however mark-recapture techniques do not work for this species, a reliable consensus graph has emerged (Figure 1). This contains 11 habitat patches (coded by N, for graph nodes) and 13 corridors (coded by L, for graph links). The names of patches are as follows: (1) "Huszas töbör", (2) "Kis tisztások", (3) "Szilicei kaszálók", (4) "U-alakú töbör", (5) "Nagy-Nyilas", (6) "Mogyorós-rét és tisztás", (7) "Árvalányhajas", (8) "Dénes töbör", (9) "Nagyoldal mögötti tisztások", (10) "Gyertyánsarjas", (11) "Lófej-forrás alatti tisztás".
Figure 1: The weighted, semi-quantitative landscape graph of the studied
Pholidoptera transsylvanica metapopulation. Nodes represent
habitat patches; the radius is proportional to local population size
quantified by {1, 2, 3, 4}. Links represent ecological corridors.
Here, thicker lines show more permeable corridors (quantified by
{1, 2, 3, 4}). Patch names (1-11) are given in text.
A semi-quantitative estimation on the quality of these spatial elements were also given. Both patch-quality (understood as local population size) and corridor-quality (understood as permeability) were quantified by integers from 1 to 4: „4” marks easily permeated corridors and large local populations. „1” marks hardly but still permeable corridors and small local populations. I refer to these semi-quantitative weights by „topography”.
3.2 Landscape graph analysis
Global indices describing the whole network and local indices describing a graph point or link and its neighbourhood are analysed. Both basic topological and topographical network properties are studied (topography now means weights on nodes and links). Figure 2 illustrates some basic considerations formalized below.
The number of neighbours of the ith node is its degree (Di). For example, the patch N3 has four neighbours (N2, N4, N5, N10), thus, DN3 = 4. For links, let degree mean the average degree of their endpoints: DL2,3 = (DN2 + DN3) / 2 = 2 + 4 / 2 = 3. A spatial elements of higher degree can be considered more important: its loss affects more seriously the connectivity of the landscape graph (i.e. it has a larger role in maintaining connectivity).
Figure 2: Six hypothetical landscape graphs illustrate how certain local indices
may reflect positional importance. The habitat patch represented by
the full circle is always in focus. Evidently, it is in more important
position (1) in "b" than in "a", because it is connected to more
neighbour patches (Db = 4 > Da = 2); (2) in "d" than in "c", because its
neighbours are less connected to each other (CCd = 0 < CCc = 0.66);
and (3) in "f" than in "e", because it is in a less periferial position
(measured by topological distance, df = 1 < de = 1.71). Further
explanation in text.
The clustering coefficient of the ith node (CCi) refers to how the neighbours of i are connected to each other: it gives the ratio of actual to possible links between these nodes in the graph. For example, node N10 has four neighbours (N3, N5, N7, N11) and only two links exist between these nodes, however the maximum number of links between four nodes equals six. Thus, CCN10 = 2/6 = 1/3. I note that it would be reasonable not to consider crossing links that cannot be drawn in two dimensions but this causes no serious failure in case of such small webs. If the clustering coefficient is smaller, the neighbours are more separated when i is deleted, thus, its importance is higher. For corridors, CCLX,Y is the average of CCNX and CCNY.
The topological distance dX,Y between two nodes X and Y of a graph is the minimal number of links connecting the two nodes. For example, the distance between nodes N7 and N9 is d7,9 = 2 (we also can go from N7 to N9 through a four-steps route including N6, but this is not a minimal route, i.e. this is not their distance). The average distance of node i in a graph (di) is its distance from a randomly selected node of the graph. For example, the distance of node N5 from others is: dN1,N5 = 3, dN2,N5 = 2, dN3,N5 = 1, dN4,N5 = 2, dN6,N5 = 1, dN7,N5 = 1, dN8,N5 = 2, dN9,N5 = 3, dN10,N5 = 1, and dN11,N5 = 2, thus dN5 = 1.8. Let the topographical distance of node i (dtgri) be a similar measure but when weights on links are considered: the length of a link does not necessarily equal 1 but five minus its weight {1, 2, 3, 4}. A highly permeable corridor („4”) can be considered just like being closer (d = 5 - 4 = 1), while a hardly permeable one („1”) behaves like a longer pathway (d = 5 - 1 = 4). So, the topographical distance of N5 from others is: dtgrN1,N5 = 5, dtgrN2,N5 = 2, dtgrN3,N5 = 1, dtgrN4,N5 = 4, dtgrN6,N5 = 4, dtgrN7,N5 = 2, dtgrN8,N5 = 3, dtgrN9,N5 = 6, dtgrN10,N5 = 2, and dtgrN11,N5 = 4, thus dtgrN5 = 3.3. For the more peripherial node N9, this is dtgrN9 = 7.4. Losing central patches (low dtgr) is more disadvantageous for connectivity. Again, for links, distance values equal the average of that of their endpoints.
In the intact network, the sum of local population size values equals 23. If certain spatial elements are deleted, the network will be separated to two or more components. One can calculate the sums of local population size values for the discrete components and it is of interest for conservationists how large is the remaining largest connected metapopulation. After deleting the ith node, the value of „maximal connected local population sizes” will be marked by LPSmaxconn(i). For example, if N3 is deleted, we will have three components: {N4}, {N1, N2}, and {N5, N6, N7, N8, N9, N10, N11}, with LPSconn values: 2, 4, and 15, respectively. So, LPSmaxconn(N3) = 15. This quantity can be calculated also for links. If LPSmaxconn is small for a spatial element, its loss is a serious damage for the whole metapopulation.
3.3 A combined importance index
For each patch and corridor, each calculated network index is given in Table 1. According to different measures of positional importance, the ranks of spatial elements differ. For example, D = 4 indicates three patches of highest importance: N3, N5, and N10. Based on CC, N3 seems to be more important to save, but dtgr suggests N5 to be of higher importance. LPS helps decision, because it strongly indicates N3. These indices refer to different local and global properties of the network, and their use is context-dependent (i.e., if the minimal viable population size is approached, the emphasis is on saving as many individuals as one can, so LPS is a more adequate index).
A combined index was constructed in order to reflect both pure topology and the quality of patches and corridors. Let the importance of the ith element of the landscape be:
, (1)
where the above introduced indices are combined according to their interrelationships. Since it has weighted components, it reflects functionality better [8, 9, 10], than some other landscape graph indices (for some more structural indices, see O’Neill et al. [11] and [12, 13, 14]). Based on this combined importance index, I give the „final” rank of landscape elements according to maintaining the connectivity of this P. transsylvanica metapopulation (Table 2).
Table 1: Four network indices of the elements of the landscape graph shown in
Figure 1. The degree (Di), clustering coefficient (CCi), and
topographical distance (dtgri) of the ith element (patch, N, and corridor,
L) are given, as well as the maximal connected local population size
value after deleting i (LPSmaxconn(i)). CC cannot be calculated for nodes
with D = 1 and links where any of the two endpoints have D = 1. Note
that different indices suggest different importance ranks. A landscape
element is considered more important in maintaining connectivity if Di
is higher, while CCi , dtgri , and LPSmaxconn(i) are lower.
i / Di / CCi / dtgri / LPSmaxconn(i)N1 / 1 / - / 7 / 20
N2 / 2 / 0 / 4,3 / 19
N3 / 4 / 0,17 / 3,6 / 15
N4 / 1 / - / 6,3 / 21
N5 / 4 / 0,33 / 3,3 / 21
N6 / 2 / 0 / 6,2 / 19
N7 / 3 / 0,33 / 4,2 / 22
N8 / 3 / 0 / 4,7 / 18
N9 / 1 / - / 7,4 / 20
N10 / 4 / 0,33 / 4,7 / 20
N11 / 1 / - / 6,5 / 22
L1/2 / 1,5 / - / 5,65 / 20
L2/3 / 3 / 0,085 / 3,95 / 19
L3/4 / 2,5 / - / 4,95 / 21
L3/5 / 4 / 0,255 / 3,45 / 23
L3/10 / 4 / 0,255 / 4,15 / 23
L5/10 / 4 / 0,33 / 4 / 23
L5/6 / 3 / 0,17 / 4,75 / 23
L5/7 / 3,5 / 0,33 / 3,75 / 23
L10/7 / 3,5 / 0,33 / 4,45 / 23
L10/11 / 2,5 / - / 5,6 / 22
L6/8 / 2,5 / 0 / 5,45 / 23
L7/8 / 3 / 0,17 / 4,45 / 23
L8/9 / 2 / - / 6,05 / 20
4 Results and conclusions
Based on our importance index, and within the limits of this approach, I suggest that the patch N3 („Szilicei kaszálók”) is of highest conservation value, at least for this seriously endangered species. Losing patch N11 („Lófej forrás alatti tisztás”) would cause the least damage to the metapopulation. Corridors, according to this analysis, are of less extreme importance. The most (L3,5) and least (L1,2) important ones differ less in importance. Of course, it must be noted that other approaches may suggest different elements to be of high conservation value – now I focused on positionality within the network.
Table 2: The combined importance index (Ii) of the ith element of the landscape
graph (patch, N, or corridor, L). The importance rank is given. Large
values of I indicate spatial elements whose loss is more